Horatio would probably tell it chronologically, beginning from the moment he met Hamlet and going through all the significant incidents, including the ghost's appearance and the final tragic end.
Yes, it can be considered a love story. Hamlet's love for Ophelia is a significant aspect. His complex feelings towards her, from his initial affection to his later erratic behavior that confounds her, are central to the play. Also, the love is intertwined with the themes of revenge, madness, and family drama, making it a very different kind of love story compared to more straightforward romantic tales.
Basically, Hamlet's story centers around Hamlet's struggle to uncover the truth about his father's death and his subsequent quest for justice. There's a lot of internal conflict and philosophical musings along the way. It's a classic tale of love, death, and revenge.